Being a vegetarian means that the grill takes on a different life. When we went shopping for a new grill 2 summers ago, my primary use case was "large surface area, as we cook pizza on the grill a lot." They wanted to sell me on BTUs, rotisseries (I think I have the rotisserie attachment in a bin in the garage somewhere...) and other whiz-bang features. I wanted square footage, or, more accurately since we're talking grill surface here, square inchage.
So to cook pizza on the grill - Start with a can of pizza dough (or use trader joe's pizza dough in a bag - 99c/each - hard to beat the price.) Can also make dough in a bread machine or...well, there's that whole "by hand" thing but that's just crazy talk. I suggest cutting into four pieces (or however many are in your family), then rolling each out. Easier to handle on the grill. Keep it on the thicker size, easier to handle. Heat up grill pretty hot (not too hot or the dough burns...) Slap them puppies, I mean dough, on the grill. Cook for 1-2 minutes until the bottom starts to get firm, and you can see grill marks. Avoid burn. Tastes bitter.
Flip onto a cookie sheet or other platter/pan/whatever you've helpfully carried out to the grill with you. You want the grilled part facing UP, uncooked side down.Dress that grilled side however you like your 'za. Kids just do plain old contadina bottled sauce and store-bought shredded mozzarella. I of course make a REAL grown-up pizza with sliced tomatoes, basil (fresh from the garden with a youngest child who is old enough to know which plant is basil, woot!) thin sliced garlic, and fresh mozzarella, because what honestly is better than that? Yum. I do a teensy bit of olive oil on the crust and some pizza sauce or jarred pasta sauce cause it's thicker and I happened to have some in the fridge.
Then put the full pizzas back on the grill to cook the underside. (Reality check: We grill the kids' pizzas first, so they're done and cooling by the time we're ready to eat.) Close the lid to let the cheese melt. it only takes a few minutes - 2-3 or less. You can even kill the flame partway through and let the heat just melt the rest of the cheese/toast the bottom a bit.
The original recipe that I stole this from included fresh cooked cut corn, which is really awesome on the fresh moz/tomato/basil one. John's tonight had asparagus on it, and regular mozzarella. Both of ours were very yummy but mine got a bit burned on the bottom, boo. Was able to eat around the burned part and stuff myself silly, have no fear. ;)